= Solution
Disintegrate $Q$ successively as
$$
Q(dy)=Q_1(dy_1)Q_2(dy_2\mid y_1)\cdots Q_N(dy_N\mid y_{<N}).
$$
For each history $y_{<i}$, choose an optimal coupling of $P_i$ and $Q_i(\cdot\mid y_{<i})$. Sampling these couplings recursively produces a joint law of $(X,Y)$ with $Y\sim Q$. Its conditional $X_i$-marginal is always $P_i$ and is independent of the past, so $X\sim P_1\otimes\cdots\otimes P_N=P$. It is therefore a coupling $\pi\in\Pi(P,Q)$.
Write $c_i(y_{<i})$ for the conditional expected cost $w(X_i,Y_i)$ in the chosen coordinate coupling. The assumed one-coordinate <transport-entropy inequality> gives
$$
\phi(c_i(y_{<i}))
\leq D\bigl(Q_i(\cdot\mid y_{<i})\Vert P_i\bigr).
$$
By <Jensen inequality> and the <chain rule for relative entropy>,
$$
\sum_{i=1}^N\phi\bigl(\mathbb E_\pi w(X_i,Y_i)\bigr)
\leq\sum_{i=1}^N\mathbb E_Q\phi(c_i(Y_{<i}))
\leq\sum_{i=1}^N\mathbb E_QD(Q_i(\cdot\mid Y_{<i})\Vert P_i)
=D(Q\Vert P).
$$
Taking the infimum over all couplings proves the <tensorization of a transport-entropy inequality>.
Solved by gpt-5.6-sol high.
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